Analysis and use of several generalized ℋ(∞ mixed sensitivity frameworks for stable multivariable plants subject to simultaneous output and input loop breaking specifications

In this paper, we present and examine three generalized mixed-sensitivity control design frameworks for linear time invariant (LTI) plants for trading off properties at distinct multivariable loop-breaking points, while being able to handle a broad class of closed loop (e.g. ℋ∞, ℋ2, frequency- and time domain) specifications. Multiobjective tradeoff paradigms are developed and analysed for ill-conditioned plants having large relative gain array entries - plants that have received considerable attention in the literature without yielding a direct systematic design methodology. We provide insight into the effectiveness of each approach and discuss the trading-off of properties at distinct loop-breaking points. This is done by exploiting the Youla-Jabr-Bongiorno-Kucera-Zames (YJBKZ) parameterization, the resulting convexification, and efficient state-of-the-art convex solvers that can be applied to smooth as well as non-differentiable problems. Moreover, we also show how our approach can be applied to multivariable infinite-dimensional plants. Specifically, by using finite dimensional approximants that converge in the uniform topology, we obtain near-optimal finite dimensional controllers for the infinite dimensional plant. Illustrative examples are provided for a thermal PDE and a retarded time delay system.

[1]  Claude Tadonki,et al.  Proximal-ACCPM: A Versatile Oracle Based Optimisation Method , 2007 .

[2]  K. Glover,et al.  Mixed H-2 and H-infinity performance objectives II: optimal control , 1994 .

[3]  W von Seelen,et al.  [Methods for comparison of the efficiency of biological and technical systems]. , 1972, Kybernetik.

[4]  Dante C. Youla,et al.  Modern Wiener-Hopf Design of Optimal Controllers. Part I , 1976 .

[5]  Eric C. Kerrigan,et al.  When is the discretization of a PDE good enough for control? , 2009, 2009 IEEE International Conference on Control and Automation.

[6]  Stephen P. Boyd,et al.  Linear controller design: limits of performance , 1991 .

[7]  Suat Gumussoy,et al.  Sensitivity minimization by stable controllers: An interpolation approach for suboptimal solutions , 2007, 2007 46th IEEE Conference on Decision and Control.

[8]  Armando A. Rodriguez,et al.  On the computation of induced norms for non-compact Hankel operators arising from distributed control problems , 1992 .

[9]  A. Nazli Gündes,et al.  Low order controller design for systems with time delays , 2011, IEEE Conference on Decision and Control and European Control Conference.

[10]  Kaustav Mondal,et al.  Fundamental control system design issues for scramjet-powered hypersonic vehicles , 2015 .

[11]  Yurii Nesterov,et al.  Homogeneous Analytic Center Cutting Plane Methods for Convex Problems and Variational Inequalities , 1999, SIAM J. Optim..

[12]  Jean-Philippe Vial,et al.  Convex nondifferentiable optimization: A survey focused on the analytic center cutting plane method , 2002, Optim. Methods Softw..

[13]  J.S. Freudenberg.  Analysis and Design for Ill-Conditioned Plants , 1988, 1988 American Control Conference.

[14]  Hans Zwart,et al.  An Introduction to Infinite-Dimensional Linear Systems Theory , 1995, Texts in Applied Mathematics.

[15]  M. V. Meerov.  Multivariable Control Systems , 1968 .

[16]  Yutaka Yamamoto,et al.  Sensitivity Reduction by Strongly Stabilizing Controllers for MIMO Distributed Parameter Systems , 2012, IEEE Transactions on Automatic Control.

[17]  Malcolm C. Smith,et al.  On the optimal two block H∞ compensators for distributed unstable plants , 1992, 1992 American Control Conference.

[18]  Andreas Griewank,et al.  On constrained optimization by adjoint based quasi-Newton methods , 2002, Optim. Methods Softw..

[19]  J.S. Freudenberg,et al.  Relations between Properties of Multivariable Feedback Systems at Different Loop-Breaking Points: Part II , 1986, 1986 American Control Conference.

[20]  Jasbir S. Arora,et al.  Survey of multi-objective optimization methods for engineering , 2004 .

[21]  Sigurd Skogestad,et al.  The use of RGA and condition number as robustness measures , 1996 .

[22]  G. Zames.  Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses , 1981 .

[23]  Vladimír Kucera.  Algebraic theory of discrete optimal control for multivariable systems [I.] , 1974, Kybernetika.

[24]  J. E. Kelley,et al.  The Cutting-Plane Method for Solving Convex Programs , 1960 .

[25]  Allen Tannenbaum,et al.  A solution to the standard H∞ problem for multivariable distributed systems , 1989 .

[26]  Carl N. Nett,et al.  The role of the condition number and the relative gain array in robustness analysis , 1994, Autom..

[27]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[28]  C. Scherer,et al.  Multiobjective output-feedback control via LMI optimization , 1997, IEEE Trans. Autom. Control..

[29]  M. A. Dahleh,et al.  Weighted H/sup infinity / optimization for stable infinite dimensional systems using finite dimensional techniques , 1990, 29th IEEE Conference on Decision and Control.

[30]  Pierre Apkarian,et al.  Mixed H2/Hinfinity Control via Nonsmooth Optimization , 2008, SIAM J. Control. Optim..

[31]  Armando A. Rodriguez,et al.  Wiener-Hopf Control of Stable Infinite Dimensional Systems , 1991, 1991 American Control Conference.

[32]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

[33]  Armando A. Rodriguez,et al.  H∞ mixed-sensitivity optimization for distributed parameter plants subject to convex constraints , 2007, 2007 46th IEEE Conference on Decision and Control.

[34]  Stephen P. Boyd,et al.  Multiobjective H/sub 2//H/sub /spl infin//-optimal control via finite dimensional Q-parametrization and linear matrix inequalities , 1998, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207).

[35]  J. Freudenberg.  Directionality, coupling, and multivariable loop-shaping , 1988, Proceedings of the 27th IEEE Conference on Decision and Control.

[36]  Karan Puttannaiah,et al.  A generalized ℋ∞ control design framework for stable multivariable plants subject to simultaneous output and input loop breaking specifications , 2015, 2015 American Control Conference (ACC).

[37]  C.D. Charalambous,et al.  A convex programming approach to the multiobjective H/sup 2//H/sup /spl infin// problem , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

[38]  Armando A. Rodriguez,et al.  Control of distributed parameter systems subject to convex constraints: Applications to irrigation systems and Hypersonic Vehicles , 2008, 2008 47th IEEE Conference on Decision and Control.

[39]  C. Scherer.  Multiobjective H/sub 2//H/sub /spl infin// control , 1995 .

[40]  Kemin Zhou,et al.  Mixed /spl Hscr//sub 2/ and /spl Hscr//sub /spl infin// performance objectives. II. Optimal control , 1994 .

[41]  André C. M. Ran.  A course in H∞− control theory , 1988 .

[42]  Onur Toker,et al.  H∞ optimal and suboptimal controllers for infinite dimensional SISO plants , 1995, IEEE Trans. Autom. Control..

[43]  Hitay Ozbay,et al.  Proper orthogonal decomposition for reduced order modeling: 2D heat flow , 2003, Proceedings of 2003 IEEE Conference on Control Applications, 2003. CCA 2003..

[44]  Pierre Apkarian,et al.  Mixed H2/H∞ control via nonsmooth optimization , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[45]  J. Freudenberg,et al.  Relations between properties of multivariable feedback systems at different loop-breaking points: Part I , 1985, 1985 24th IEEE Conference on Decision and Control.

[46]  H∞ sensitivity minimization for unstable infinite-dimensional plants , 1993, 1993 American Control Conference.

[47]  Karan Puttannaiah.  H-Infinity Control Design Via Convex Optimization: Toward A Comprehensive Design Environment , 2013 .

[48]  Armando A. Rodriguez.  Weighted H∞ mixed-sensitivity minimization for stable MIMO distributed-parameter plants , 1995 .

[49]  M. J. Khosrowjerdi,et al.  Multiobjective H2/H∞ control design for a VSTOL flight model , 2010, 2010 18th Iranian Conference on Electrical Engineering.

[50]  O. Cifdaloz,et al.  Constrained H~~ Mixed-Sensitivity Optimization for Stable Infinite-Dimensional Plants: Application to Thermal Diffusion Process , 2006, 2006 American Control Conference.

[51]  Ian Postlethwaite,et al.  Multivariable Feedback Control: Analysis and Design , 1996 .

[52]  D. Peaucelle,et al.  Robust Multi-Objective Control toolbox , 2006, 2006 IEEE Conference on Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control.

[53]  O. Cifdaloz,et al.  H/sup /spl infin// mixed sensitivity minimization for stable infinite-dimensional plants subject to convex constraints , 2005, Proceedings of the 2005, American Control Conference, 2005..

[54]  Munther A. Dahleh,et al.  Weighted ?Im Optimization for Stable Infinite Dimensional Systems using Finite Dimensional Techniques * , 1990 .

[55]  J. Dennis,et al.  A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems , 1997 .

[56]  O. Cifdaloz,et al.  Constrained H∞ Mixed-Sensitivity Optimization for Infinite-Dimensional Plants: Applications to Thermal, Structural, and Aircraft Systems , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[57]  Oguzhan Cifdaloz.  H -infinity mixed-sensitivity optimization for infinite dimensional plants subject to convex constraints , 2007 .